Metrization Theorem for Uniform Loops with the Invertibility Property

Dagmar Markechová 1,*, Peter Vrábel 1 and Beáta Stehlíková 2 1 Department of Mathematics, Faculty of Natural Sciences, Constantine the Philosopher University in Nitra, A. Hlinku 1, SK-949 74 Nitra, Slovakia; pvrabel@ukf.sk 2 Department of Informatics and Mathematics, Faculty of Economy and Business, Pan European University, SK-851 05 Bratislava, Slovakia; stehlikovab@gmail.com * Correspondence: dmarkechova@ukf.sk; Tel.: +421-376-408-111; Fax: +421-376-408-020


Introduction
The theory of topological groups [1][2][3][4][5][6] is a currently well-established field.In contrast, the theory of topological quasigroups and loops [7][8][9][10][11][12] is not as well developed.This is because many of the constructions and conditions one takes for granted in the former theory do not extend easily to the latter.In our recently published papers [13,14], we dealt with the study of uniform loops with the invertibility property (uniform IP-loops, for short).In [13] we proved that in every locally compact topological IP-loop with topology induced by a left-invariant uniformity there exists at least one regular left Haar measure and in [14] we proved that this measure is essentially unique.
In this contribution, we extend our study concerning uniform IP-loops.Our aim is to provide a metrization theorem for uniform IP-loops.A metrization theorem for topological groups is proved in [2] (see also [1,3]).In this theorem, the sufficient conditions for a topological group X to be metrizable by a left-invariant metric are given.Our purpose in this paper is to prove an analogy of this metrizable theorem for the case that X does not have a group structure, X is only a quasigroup.A topological quasigroup is not metrizable by a left-invariant metric in general what we illustrate by presented example.IP-loops are a special case of quasigroups.We give here the conditions ensuring that an IP-loop with a left-invariant uniform topology is metrizable by a left-invariant metric.By consideration of the topological IP-loop X dual to X we obtain that an IP-loop with a right-invariant uniform topology is metrizable by a right-invariant metric.

Basic Definitions and Facts
First, we recall the definitions of basic notions and some known facts which will be used in the following.
A quasigroup is a groupoid (X, •) in which for every two elements a, b ∈ X every of the equations ax = b and ya = b has a unique solution in X.Instead of a • b we write ab.If a quasigroup X contains an identity element, then X is called a loop.Evidently, every associative loop is a group.A quasigroup (X, •) is called an IP-quasigroup (or a quasigroup with the invertibility property), if there exist mappings f P : X → X and f L : X → X such that, for any x, y ∈ X, it holds (xy) f P (y) = x and f L (x)(xy) = y.An IP-quasigroup with an identity element is called an IP-loop.In any IP-loop every element x in X has an inverse x −1 ∈ X and it holds (xy) −1 = y −1 x −1 for every x, y ∈ X.It is easy to see that IP-loop is a groupoid (X, •) with an identity element and with the following property: for each x ∈ X there exists an element x −1 ∈ X such that (yx)x −1 = y and x −1 (xy) = y for every y ∈ X.
The conjugate of an octonion: x = x 0 +x 1 e 1 +x 2 e 2 +x 3 e 3 +x 4 e 4 +x 5 e 5 +x 6 e 6 +x 7 e 7 (2) is given by: x * = x 0 −x 1 e 1 −x 2 e 2 −x 3 e 3 −x 4 e 4 −x 5 e 5 −x 6 e 6 −x 7 e 7 . ( Conjugation is an involution of O and satisfies (xy) * = y * x * .This octonionic multiplication is neither commutative (e i e j = −e j e i if i, j are distinct and non-zero) nor associative.On other hand, the nonzero elements of O form an IP-loop.The norm of the octonion x is defined as: where x * is the conjugate of x.So this norm agrees with standard Euclidean norm on 8 .The existence of a norm on O implies the existence of inverses for all nonzero elements of O.The inverse of x, x = 0, is given by the equality x −1 = x * x 2 .It satisfies xx −1 = x −1 x = 1.The couple (O n , •), where the operation • is defined by the equality: is also an IP-loop.
A metric d on a groupoid (X, •) is said to be left-invariant, if d(ax, ay) = d(x, y) for every a, x, y ∈ X.A right-invariant metric is defined analogously and a metric is said to be bi-invariant if it is both left and right invariant.The term "invariant" hence means that the distance is unchanged when you translate by a fixed element a.If X is abelian, then both left and right invariance implies bi-invariance and we simply say that d is invariant.

Example 1.
Let O 1 be a set of all octonions with a unit norm.Let d be the metric induced by the norm on O 1 .Let x, y, a be octonions with a unit norm: the metric d is left-invariant.Analogously we obtain that d is right-invariant.
where O 1 is a set of octonions with a unit norm, x = (x 1 , . . ., x n ), y = (y 1 , . . ., y n ), a = (a 1 , . . ., a n ).Put: where d is the metric from Example 1.It is easy to verify that d is also a metric.Since d is left-invariant, we obtain: This means that the metric d is left-invariant.Analogously we obtain that the metric d is right-invariant.
A topological IP-loop is an IP-loop (X, •) with Hausdorff topology such that the transformation X × X → X defined by (x, y) → x −1 y is continuous.A topological IP-loop is said to be connected, totally disconnected, compact, locally compact, etc., if the corresponding property holds for its underlying topological space.Topological IP-loops are studied, e.g., in [18][19][20][21].
In the following section, we will deal with topological IP-loops with a left-invariant uniformity.Therefore, we recall the definition of a uniform topology and remind the facts which will be further used.
A uniformity of a set X is non-empty system W of subsets of the Cartesian product X × X such that the following conditions are satisfied: The above described couple (X, W) is called a uniform space and its elements entourages.Every uniform space X becomes a topological space by defining a subset U of X to be open if and only if for every x ∈ U there exists an entourage V such that V[x] = {y ∈ X; (x, y) ∈ V} is a subset of U. A basis of a uniformity W is any system Mathematics 2017, 5, 32 3 of 8 Since: where d is the metric from Example 1.It is easy to verify that d is also a metric.Since d is left-invariant, we obtain: ).

, ( y x d
This means that the metric d is left-invariant.Analogously we obtain that the metric d is right-invariant. with Hausdorff topology such that the transformation X  X X defined by (x, y)  x -1 y is continuous.A topological IP-loop is said to be connected, totally disconnected, compact, locally compact, etc., if the corresponding property holds for its underlying topological space.Topological IP-loops are studied, e.g., in [18][19][20][21].
In the following section, we will deal with topological IP-loops with a left-invariant uniformity.Therefore, we recall the definition of a uniform topology and remind the facts which will be further used.
A uniformity of a set X is non-empty system W of subsets of the Cartesian product X  X such that the following conditions are satisfied: (1) Every element of W contains the diagonal }. ); , {( The above described couple ) , ( W X is called a uniform space and its elements entourages.
Every uniform space X becomes a topological space by defining a subset U of X to be open if and only if for every A basis of a uniformity W is any system Β of entourages of W such that every entourage of W contains a set belonging to Β .Every uniform space has a basis of entourages consisting of symmetric entourages.It is known (see [22,23]) that any system Β of subsets of the Cartesian product X X  is a basis of some uniformity of X if and only if the following conditions are satisfied:  with Hausdorff topology such that the ansformation X  X X defined by (x, y)  x -1 y is continuous.A topological IP-loop is said to be onnected, totally disconnected, compact, locally compact, etc., if the corresponding property holds r its underlying topological space.Topological IP-loops are studied, e.g., in [18][19][20][21].
In the following section, we will deal with topological IP-loops with a left-invariant uniformity.herefore, we recall the definition of a uniform topology and remind the facts which will be further used.
A uniformity of a set X is non-empty system W of subsets of the Cartesian product X  X such at the following conditions are satisfied: (1) Every element of W contains the diagonal }. ); , {( is called a uniform space and its elements entourages.
very uniform space X becomes a topological space by defining a subset U of X to be open if and nly if for every is a subset of .A basis of a uniformity W is any system Β of entourages of W such that every entourage of W ontains a set belonging to Β . Every uniform space has a basis of entourages consisting of symmetric ntourages.It is known (see [22,23]) that any system Β of subsets of the Cartesian product X X  a basis of some uniformity of X if and only if the following conditions are satisfied: . Every uniform space has a basis of entourages consisting of symmetric entourages.It is known (see [22,23]) that any system Mathematics 2017, 5, 32 3 of 8 Since: O is a set of octonions with a unit norm, ), ,..., ( ).

, ( y x d
This means that the metric d is left-invariant.Analogously we obtain that the metric d is right-invariant.
A topological IP-loop is an IP-loop ) , (  X with Hausdorff topology such that the transformation X  X X defined by (x, y)  x -1 y is continuous.A topological IP-loop is said to be connected, totally disconnected, compact, locally compact, etc., if the corresponding property holds for its underlying topological space.Topological IP-loops are studied, e.g., in [18][19][20][21].
In the following section, we will deal with topological IP-loops with a left-invariant uniformity.Therefore, we recall the definition of a uniform topology and remind the facts which will be further used.
A uniformity of a set X is non-empty system W of subsets of the Cartesian product X  X such that the following conditions are satisfied: is called a uniform space and its elements entourages.
Every uniform space X becomes a topological space by defining a subset U of X to be open if and only if for every A basis of a uniformity W is any system Β of entourages of W such that every entourage of W contains a set belonging to Β . Every uniform space has a basis of entourages consisting of symmetric entourages.It is known (see [22,23]) that any system Β of subsets of the Cartesian product X X  is a basis of some uniformity of X if and only if the following conditions are satisfied: of subsets of the Cartesian product X × X is a basis of some uniformity of X if and only if the following conditions are satisfied: (6) ). , ( y x d is means that the metric d is left-invariant.Analogously we obtain that the metric d is right-invariant.
A topological IP-loop is an IP-loop ) , (  X with Hausdorff topology such that the nsformation X  X X defined by (x, y)  x -1 y is continuous.A topological IP-loop is said to be nected, totally disconnected, compact, locally compact, etc., if the corresponding property holds its underlying topological space.Topological IP-loops are studied, e.g., in [18][19][20][21].
In the following section, we will deal with topological IP-loops with a left-invariant uniformity.erefore, we recall the definition of a uniform topology and remind the facts which will be further used.
A uniformity of a set X is non-empty system W of subsets of the Cartesian product X  X such t the following conditions are satisfied: The above described couple ) , ( W X is called a uniform space and its elements entourages.
ery uniform space X becomes a topological space by defining a subset U of X to be open if and ly if for every A basis of a uniformity W is any system Β of entourages of W such that every entourage of W tains a set belonging to Β . Every uniform space has a basis of entourages consisting of symmetric tourages.It is known (see [22,23]) that any system Β of subsets of the Cartesian product X X  basis of some uniformity of X if and only if the following conditions are satisfied: O is a set of octonions with a unit norm, ), ,..., ( ). , ( y x d t the metric d is left-invariant.Analogously we obtain that the metric d is right-invariant.
ogical IP-loop is an IP-loop ) , (  X with Hausdorff topology such that the n X  X X defined by (x, y)  x -1 y is continuous.A topological IP-loop is said to be tally disconnected, compact, locally compact, etc., if the corresponding property holds ying topological space.Topological IP-loops are studied, e.g., in [18][19][20][21].llowing section, we will deal with topological IP-loops with a left-invariant uniformity.recall the definition of a uniform topology and remind the facts which will be further used.mity of a set X is non-empty system W of subsets of the Cartesian product X  X such ing conditions are satisfied: is called a uniform space and its elements entourages.
space X becomes a topological space by defining a subset U of X to be open if and ry is a subset of a uniformity W is any system Β of entourages of W such that every entourage of W belonging to Β . Every uniform space has a basis of entourages consisting of symmetric t is known (see [22,23]) that any system Β of subsets of the Cartesian product X X  me uniformity of X if and only if the following conditions are satisfied: , then there exists V ∈ Mathematics 2017, 5, 32 3 of 8 Since: ).

, ( y x d
This means that the metric d is left-invariant.Analogously we obtain that the metric d is right-invariant.
A topological IP-loop is an IP-loop ) , (  X with Hausdorff topology such that the transformation X  X X defined by (x, y)  x -1 y is continuous.A topological IP-loop is said to be connected, totally disconnected, compact, locally compact, etc., if the corresponding property holds for its underlying topological space.Topological IP-loops are studied, e.g., in [18][19][20][21].
In the following section, we will deal with topological IP-loops with a left-invariant uniformity.Therefore, we recall the definition of a uniform topology and remind the facts which will be further used.
A uniformity of a set X is non-empty system W of subsets of the Cartesian product X  X such that the following conditions are satisfied: (1) Every element of W contains the diagonal }. ); , {( The above described couple ) , ( W X is called a uniform space and its elements entourages.
Every uniform space X becomes a topological space by defining a subset U of X to be open if and only if for every A basis of a uniformity W is any system Β of entourages of W such that every entourage of W contains a set belonging to Β . Every uniform space has a basis of entourages consisting of symmetric entourages.It is known (see [22,23]) that any system Β of subsets of the Cartesian product X X  is a basis of some uniformity of X if and only if the following conditions are satisfied: with Hausdorff topology such that the n X  X X defined by (x, y)  x -1 y is continuous.A topological IP-loop is said to be tally disconnected, compact, locally compact, etc., if the corresponding property holds ying topological space.Topological IP-loops are studied, e.g., in [18][19][20][21].llowing section, we will deal with topological IP-loops with a left-invariant uniformity.recall the definition of a uniform topology and remind the facts which will be further used.mity of a set X is non-empty system W of subsets of the Cartesian product X  X such ing conditions are satisfied: is called a uniform space and its elements entourages.
space X becomes a topological space by defining a subset U of X to be open if and ry is a subset of a uniformity W is any system Β of entourages of W such that every entourage of W belonging to Β . Every uniform space has a basis of entourages consisting of symmetric t is known (see [22,23]) that any system Β of subsets of the Cartesian product X X  me uniformity of X if and only if the following conditions are satisfied: , then there exists V ∈ Mathematics 2017, 5, 32 3 of 8 Since:  ).

, ( y x d
This means that the metric d is left-invariant.Analogously we obtain that the metric d is right-invariant.
A topological IP-loop is an IP-loop ) , (  X with Hausdorff topology such that the transformation X  X X defined by (x, y)  x -1 y is continuous.A topological IP-loop is said to be connected, totally disconnected, compact, locally compact, etc., if the corresponding property holds for its underlying topological space.Topological IP-loops are studied, e.g., in [18][19][20][21].
In the following section, we will deal with topological IP-loops with a left-invariant uniformity.Therefore, we recall the definition of a uniform topology and remind the facts which will be further used.
A uniformity of a set X is non-empty system W of subsets of the Cartesian product X  X such that the following conditions are satisfied: (1) Every element of W contains the diagonal }. ); , {( The above described couple ) , ( W X is called a uniform space and its elements entourages.
Every uniform space X becomes a topological space by defining a subset U of X to be open if and only if for every A basis of a uniformity W is any system Β of entourages of W such that every entourage of W contains a set belonging to Β . Every uniform space has a basis of entourages consisting of symmetric entourages.It is known (see [22,23]) that any system Β of subsets of the Cartesian product X X  is a basis of some uniformity of X if and only if the following conditions are satisfied: O is a set of octonions with a unit norm, ), ,..., ( ). , ( y x d that the metric d is left-invariant.Analogously we obtain that the metric d is right-invariant. ological IP-loop is an IP-loop ) , (  X with Hausdorff topology such that the tion X  X X defined by (x, y)  x -1 y is continuous.A topological IP-loop is said to be totally disconnected, compact, locally compact, etc., if the corresponding property holds erlying topological space.Topological IP-loops are studied, e.g., in [18][19][20][21].following section, we will deal with topological IP-loops with a left-invariant uniformity.we recall the definition of a uniform topology and remind the facts which will be further used.formity of a set X is non-empty system W of subsets of the Cartesian product X  X such lowing conditions are satisfied: is called a uniform space and its elements entourages.
orm space X becomes a topological space by defining a subset U of X to be open if and every is a subset of of a uniformity W is any system Β of entourages of W such that every entourage of W set belonging to Β . Every uniform space has a basis of entourages consisting of symmetric .It is known (see [22,23]) that any system Β of subsets of the Cartesian product X X  f some uniformity of X if and only if the following conditions are satisfied: , then there exists M ∈ Mathematics 2017, 5, 32 3 of 8 Since: O is a set of octonions with a unit norm, ), ,..., ( ).

, ( y x d
This means that the metric d is left-invariant.Analogously we obtain that the metric d is right-invariant. A topological IP-loop is an IP-loop ) , (  X with Hausdorff topology such that the transformation X  X X defined by (x, y)  x -1 y is continuous.A topological IP-loop is said to be connected, totally disconnected, compact, locally compact, etc., if the corresponding property holds for its underlying topological space.Topological IP-loops are studied, e.g., in [18][19][20][21].
In the following section, we will deal with topological IP-loops with a left-invariant uniformity.Therefore, we recall the definition of a uniform topology and remind the facts which will be further used.
A uniformity of a set X is non-empty system W of subsets of the Cartesian product X  X such that the following conditions are satisfied: The above described couple ) , ( W X is called a uniform space and its elements entourages.
Every uniform space X becomes a topological space by defining a subset U of X to be open if and only if for every A basis of a uniformity W is any system Β of entourages of W such that every entourage of W contains a set belonging to Β . Every uniform space has a basis of entourages consisting of symmetric entourages.It is known (see [22,23]) that any system Β of subsets of the Cartesian product X X  is a basis of some uniformity of X if and only if the following conditions are satisfied: Example 3. Let be the set of all real numbers and • be a binary operation on defined, for every a, b ∈ , by the prescription a • b = a+b 2 .Evidently, the couple ( , •) is a topological quasigroup with standard topology.We shall show that this topological quasigroup is not metrizable topological space by an invariant metric.Suppose that there is a left-invariant metric d in ( , •) and the metric topology induced by d coincides with standard topology.So the metric d and standard metric on are equivalent.We will derive a contradiction.Let x, y ∈ , x = y.Since d is left-invariant, for every element a in , we have: This means that the topological quasigroup ( , •) with standard topology is not metrizable topological space by left-invariant metric, and also by right-invariant metric because the operation • is commutative.
Proof.Let us define a non-negative real-valued function f : X × X → as follows: Then the searched function d has the following form: where the infimum is taken over all elements x 1 , x 2 , . . ., x n ∈ X.
We shall prove that the function d has the required properties.For every x, y, z ∈ X, we have: This means that the first property holds.Now, we shall prove the second property.Since (X, •) is an IP-loop, the condition (a, a)U n = U n is equivalent to the condition (a −1 , a −1 )U n = U n , for n = 0, 1, 2, . . ., and every a ∈ X. Therefore (x, y) ∈ U n−1 − U n if and only if (x, y) ∈ (a −1 , a −1 )U n−1 − (a −1 , a −1 )U n .Thus, (x, y) ∈ U n−1 − U n if and only if (ax, ay) ∈ U n−1 − U n . Since: ax, ay) ∈ U n for every n, we get f (ax, ay) = f (x, y), for every a, x, y ∈ X.
It remains to prove the third property.The inclusions U n+1 ⊂ U n , n = 0, 1, 2, . . ., are the consequence of the previous two conditions.If (x, y) ∈ U n , then (x, y) ∈ U k , k = 0, 1, . . ., n, and Let us prove the inclusion {(x, y); d(x, y) < 2 −n } ⊂ U n−1 , for n = 1, 2, . . . .If d(x, y) < 2 −n , then there are the elements x 0 , x 1 , . . ., x k , x k+1 , where x 0 = x and x k+1 = y, such that: By induction on k we prove that (x 0 , x k+1 ) Assume that it holds for some k ≥ 1, i.e., f (z 0 , z Since, by means of the second condition, we have (x, x) ∈ U n , for n = 0, 1, 2, . . ., it holds f (x, x) = 0, for every x ∈ X, and so d(x, x) = 0, for every x ∈ X.If {U n } ∞ n=0 is a sequence of symmetric subsets of X × X, then f (x, y) = f (y, x), for every x, y ∈ X, and therefore d(x, y) = d(y, x) , for every x, y ∈ X.The constructed function d is a pseudometric.The proof is completed.Theorem 1.Let (X, •) be an IP-loop.A uniform space (X, W) is pseudometrizable by a left-invariant pseudometric if and only if the uniformity W of a set X has a left-invariant countable base.A uniform space (X, W) is metrizable by a left-invariant metric if and only if the corresponding topological space is Hausdorff and the uniformity W of X has a left-invariant countable base.
Proof.If a uniformity W of a set X has a left-invariant countable base, then we can construct a system of symmetric subsets of a set X × X satisfying the condition of the preceding proposition.Therefore, the uniform space (X, W) is pseudometrizable by a left-invariant pseudometric.The second part of assertion of the theorem is obvious.
If (X, •) is any topological IP-loop, one can consider the topological IP-loop ( X, •) dual to X.The topological IP-loop X has, by definition, the same elements and same topology as X,the product • in X is defined , for every x, y ∈ X, by x • y = y • x.Since, for every x, y ∈ X, it holds: we see that ( X, •) is in fact an IP-loop.Let (X, •) be a right-invariant uniform IP-loop.Consider the topological IP-loop ( X, •) dual to X.If  op is an IP-loop ) , (  X with Hausdorff topology such that the defined by (x, y)  x -1 y is continuous.A topological IP-loop is said to be ected, compact, locally compact, etc., if the corresponding property holds ical space.Topological IP-loops are studied, e.g., in [18][19][20][21].on, we will deal with topological IP-loops with a left-invariant uniformity.inition of a uniform topology and remind the facts which will be further used.
X is non-empty system W of subsets of the is subset of W is any system Β of entourages of W such that every entourage of W Β .Every uniform space has a basis of entourages consisting of symmetric ee [22,23]) that any system Β of subsets of the Cartesian product X X  ity of X if and only if the following conditions are satisfied: ).

, ( y x d
This means that the metric d is left-invariant.Analogously we obtain that the metric d is right-invariant. A topological IP-loop is an IP-loop ) , (  X with Hausdorff topology such that the transformation X  X X defined by (x, y)  x -1 y is continuous.A topological IP-loop is said to be connected, totally disconnected, compact, locally compact, etc., if the corresponding property holds for its underlying topological space.Topological IP-loops are studied, e.g., in [18][19][20][21].
In the following section, we will deal with topological IP-loops with a left-invariant uniformity.Therefore, we recall the definition of a uniform topology and remind the facts which will be further used.
of entourages of W such that every entourage of W contains a set belonging to athematics 2017, where d is the metric from Example 1.It is easy to verify that d is also a metric.Since d is left-invariant, we obtain: j , y j+1 ); y 1 , . . ., y m ∈ X, y 0 = y, y m+1 = z = d(x, y) + d(y, z).
we obtain that d is right-invariant.

1 O 1 .
is a set of octonions with a unit norm, It is easy to verify that d is also a metric.Since d is left-invariant, left-invariant.Analogously we obtain that the metric d is right-invariant.

1 O
is a right-invariant base of the uniformity W of (X, •), then Mathematics 2017, is left-invariant.Analogously we obtain that d is right-invariant.is a set of octonions with a unit norm, the metric from Example 1.It is easy to verify that d is also a metric.Since d is left-invariant, we obtain: 2 e 2 + x 3 e 3 + x 4 e 4 + x 5 e 5 + x 6 e 6 + x 7 e 7 , y = y 0 + y 1 e 1 + y 2 e 2 + y 3 e 3 + y 4 e 4 + y 5 e 5 + y 6 e 6 + y 7 e 7 , a = a 0 + a 1 e 1 + a 2 e 2 + a 3 e 3 + a 4 e 4 + a 5 e 5 + a 6 e 6 + a 7 e 7 .
Analogously we obtain that d is right-invariant.
Every element of Analogously we obtain that d is right-invariant.
Cartesian product X  X such comes a topological space by defining a subset U of X to be open if and ere exists an entourage V such